Sunday, October 19, 2014

Surfacing and studying studying misconceptions via Talking Points

In a class of 36 students, where it can be, shall we say, difficult for me to do formative assessment on every student every day, the Talking Points structure give me a great way to surface and deal with student misconceptions by getting students to surface, discuss, and correct them.

Par example...

Our sequencing for Geometry has gotten totally screwy this year because of some new district requirements around the Common Core. Having learned how to do all of the basic constructions, we are now finally approaching the unit test on parallel lines and their angles. There are so many possible errors in understanding that can happen around these, I wanted to create a group work activity to address them. I also want to change groups up this week, so I am using this as community-building as well.

I've discovered it is a good idea to weave together community-building statements, growth mindset statements, metacognitive self-monitoring statements, and Always / Sometimes / Never statements. The Talking Points structure lets me accomplish multiple goals simultaneously, which is something I need to do with such big classes.

An editable version of this set of Talking Points is on the Group Work Working Group (#GWWG14) wiki page on the TMC14 wiki.

Thursday, October 9, 2014

Constructions Castles

Einstein was right — imagination IS more important than knowledge.

I used to think of that quote a lot as I walked the Princeton campus, through quadrangles and past trees that — legend had it — he had crashed into on his bicycle while he was lost in his thoughts.

I passed his house on Mercer Street every week on my way to music lessons, and I wondered if his sister had not had the door painted it bright red so that he might notice it and not bump into it.

When I realized I would have to teach both constructions AND proof this year in Geometry, I first thought of them as a Big Obstacle. But seven weeks in and I have fallen in love with constructions. I created this Constructions Castle project to give students plenty of practice doing constructions while also giving them a chance to develop their understanding of how shapes and angles fit together.

I think their work speaks for itself.

Files are on the Math Teacher Wiki (task cardconstructions checklist, and rubric).

Friday, October 3, 2014

Formative Assessment PLUS Practice Activity: "Lightning Rounds" Review With Whiteboards

I've been looking for a way to do formative assessment that also doubles as a practice activity, and I seem to have stumbled on something good with my new "Lightning Rounds Review With Whiteboards activity. It uses some of what I appreciate most from both Steve Leinwand and Dylan Wiliam.

We have large-ish whiteboards — enough for one per table group — in our classrooms. Since I've also got around 36 students in each of my classes, I've been drowning with my older methods of checking for understanding. With 175 total students, even a 1-minute-per-exit-ticket assessment can take ages to go through — plus all the time it takes me to make it up.

This activity has been working much better. It also takes less prep up front and allows me to engage more deeply with the students and/or groups who need more attention. Assessing 9 tables' work at a glance is much more manageable and less exhausting than trying to flip through 36 pieces of student work. And of course, that is better for everybody.

Here is how it worked today.

In Precalculus, we're working on evaluating trig functions of real numbers on the unit circle. Students and groups are strongly encouraged to use their unit circles and notes to help them gain confidence and fluency.

Each table group gets a board, three markers, and an eraser. On the document camera, covering the non-current rows with a folder, I reveal one row of problems, which contains three problems: (a), (b), and (c).

Students discuss and work the problems, and when they are done, they hold up their table's whiteboard for a "check-in."

I glance around the room and check three answers at a time, calling out, "Table 2, you are checked in and correct!" "Table 6, you are checked in and correct!" "Table 1, you need to reexamine part (b)."

Because there are three problems in each "round," there's plenty of time for groups to make an error, assess their work and reconsider, and re-present their findings. Because there are only nine tables reporting in, it is manageable from both a teacher and a student perspective.

Plus, of course, since there is a group whiteboard and markers to play with, those groups who finished quickly and received their checkpoint are happy to doodle or play tic-tac-toe or offer funny editorial comments or cartoons while other groups receive some attention or extra time.

Last night was our Back To School Night, and student clubs were selling food as fundraisers for their clubs. Two of my students discovered that my classes' parents and I were helpless in the face of their delicious goodies. At one point, I had told them, Look, I don't have cash; my purse is locked upstairs in my desk.

Around the 6th or 7th round, Table 7 included an editorial comment on the side of their board: "We heart Dr. S! Also, you owe us $2. :)" While everybody started working the next round of problems, I unlocked my purse out of my closet and paid my debts. Some good laughs were had by all, and it was a nice piece of mathematical community-building.

The files I used are on the Math Teacher Wiki.

All in all, an easy, effective practice activity, with embedded formative assessment and community-building built right in. Not bad for a too-hot Friday!

Sunday, September 21, 2014

Talking Points and Classroom Community

I am still drowning in papers to grade, but I wanted to write a quick post before the week got too far away from me. I had my first-ever bomb threat this past week, and it rattled us all.  I grabbed my iPhone, my MacBook Pro, and my math pencil and escorted my class out of the building.

I had to make my way home with no keys, no purse, and no money. Just carrying my iPhone and my laptop and my trusty math pencil.

As I battled public transit to make my way home (with no money – just explaining where I'd been and what had happened), it occurred to me that everybody was going to be nuts the next day. So I started writing some new Talking Points for us to do about the events of the previous day.

One of the things that becomes clear with training and time is that as a teacher, my job is to provide an emotional "container" for the experiences we have in my classroom. I decided to use Talking Points as a way of enabling students to process their experience within the structure we've been building for several weeks now.

My students' responses and respect for the structure blew my mind. And they made it possible for us to lose only one day rather than several. And by Thursday, our classroom community had returned to a steady enough state to move ahead.

Sunday, September 7, 2014

GEOMETRY – a Jane Schaffer-ish approach to teaching students' first two-column proof

I'm required to teach two-column proofs in Geometry, but having also been trained as an English teacher, this has never seemed like a problem to me. In fact, if anything, the activity I used on Friday seemed to scaffold the process using students' existing knowledge better than anything else I've tried before.

My design process was as follows: because of their ELA and writing backgrounds, students already know far more about constructing an argument in words and statements than we math teachers often give them credit for knowing. All the major writing curricula, such as Jane Schaffer and Six Traits, provide scaffolded methods for teaching students to make claims and to support their assertions with evidence and interpretations that connect that evidence to their claims through interpretive statements. Indeed, the Jane Schaffer method, in particular, has a very lovely scaffolded process (which I've extended in the past) to bridge students' metacognitive processes about their writing, taking them from a place of very concrete thinking to one of considerable abstraction.

 So why not use this same kind of process for proof?

Instead of having students merely "fill in" the reasons for the statements in their first proof (in our curriculum, that's the Midpoint Theorem), I created a task card with instructions and materials for creating a "working poster" (an idea I have adopted from Malcolm Swan) of a two-column proof. They needed to set it up the way we'd done it the day before (two columns, Statements and Reasons), and then they would need to (a) sort their cut-out statements from the task card into a correct order (more than one order is possible), and (b) use their notes and discussions to give the justification or reason that permitted them to make each of these assertions in turn.

The richness of their conversations blew me away. They also confirmed my intuitions that (1) math conversations and projects can indeed draw on students' existing competencies in argumentation that they have developed in their English and Social Studies classes (indeed, many relished the opportunity!), and (b) it is indeed possible to create intellectual need (see Guershon Harel and Dan Meyer) for definitions, postulates, previous theorems, and propositions from algebra through situational motivation.

This activity turned two-column proof into a reasoning and sense-making activity that exposed and built on prior knowledge instead of invalidating it; created what Swan calls "realistic obstacles to be overcome"; turned students' notes into a valued and valuable learning resource; and used higher-order questioning, as opposed to mere recall.
I realize I have not referenced the van Hiele levels here, but that is, in part, because I think I may be kind of bypassing some of their assumptions. I'm not at all sure about this, though, and I would welcome better-informed thoughts and thinking about this in the comments.


Task card for intro to two-column proof: 1-5 intro task Sorti…o 2-Column Proof.pdf

Editable Word doc: 1-5 intro task Sorti…o 2-Column Proof.doc

Original editable Pages doc: 1-5 intro task Sorti…2-Column Proof.pages

Monday, September 1, 2014

What do you do after Formative Assessment reveals a gaping hole in understanding? More Talking Points, of course. :)

My Geometers took the opportunity to inform me through their Chapter 1 exams that they really don't get how angles are named. So this seemed like a perfect opportunity for more Talking Points, of course. :)

This time I'm giving everybody a diagram of a figure that the Talking Points refer to. They will have to do some reasoning about naming angles in order to do the Talking Points. They love doing Talking Points, but they mostly like coming to immediate consensus. Hopefully this will throw a monkey wrench (so to speak) into those works.

Here is the Talking Points file (they print 2-UP) and here is the set of diagrams (they print 6-UP) to use together for this lesson/activity.

More news as it happens!

Sunday, August 31, 2014

PRECALCULUS: Transformations of Functions Speed Dating

I know I am going to take some grief for this statement, but I have learned it to be the truth. There are some things that, once discovered, you simply need to memorize — and practice. This is about mental (and often physical) motor skills. Becoming fluent at some things requires more or less practice for different people, but for most of us, it does require something.

I come by this knowledge honestly.

I have played the piano since I was three or four. In growing as a pianist, there are conceptual learnings, procedural learnings, and contextual learnings. Technique, concepts, and contexts. To play well, you need first to become proficient and fluent technically. That is why there are so many established sets of technical figure exercises: Hanon, Czerny, and many others. The same is true for the violin (Schradieck, Kreutzer, Hrimaly) and every other instrument under the sun. If you want to be capable of playing the advanced works in your instrument's repertoire, your fingers and body need to know how to flow over the keys or strings in every known and familiar situation (scales, arpeggios, finger-crossings) so that you can summon them automatically in your pursuit of unknown and unfamiliar territory in the literature of your instrument.

But working your way through all of the possible finger movements will only make you a master technician. Doing only finger exercises will not give you a very deep understanding of music. It simply builds the finger and muscle memory you will need to face different and difficult technical situations and requirements as you dive deeper and deeper into the piano repertoire.

In other words, technical proficiency is necessary, but not sufficient to become a strong and capable musician. Without technique, you cannot progress.

Beyond technique, there are concepts to master — harmony, counterpoint, and composition, as well as music history. If you don't understand the basics of scales and harmonies and chord progressions in Western music, you will lack the basic musicianship that is required to make sense of the piano repertoire. You need to understand how melodic lines or voices can be woven together using harmonies and rhythms and chord progressions to build a piece of music with coherent and reproducible grammar and syntax that can be both encoded and decoded by others. Without the foundational concepts, your attempts to interpret and play the repertoire will be incoherent.

Finally, there are contexts — historical, cultural, interpretive. The ultimate context is your instrument's repertoire. Music, like mathematics, is a cultural act. As a musical learner, you need to be mentored into the repertoire. You need to experience other musicians' interpretations and experiences to learn what competent, coherent, and ultimately subtle musical communication sounds, feels, and looks like. And while you are doing this, you also need to be able to explore the repertoire for yourself so that you can find your own way.

It seems to me that this is a similar situation to what we face in mathematics education. Technical proficiency is boring but somewhat mindless. If you had to listen to anyone (including yourself) only playing Hanon's same 60 exercises day in and day out, you would undoubtedly lose your mind. As music, the technical patterns are boring — up and down, back and forth, crossing and uncrossing, stretching and shifting. But they're necessary to develop a foundation of muscle memory and motor skills, as well as the habits of mind and of practice you will need as you gain proficiency and advance to building the finer and finer skills of musicianship.

So there was a need for teachers to create pieces that are very rich musically but very restricted technically so that they are accessible to new learners. This is why Bach, for example, wrote the pieces in his Anna Magdalena Notebook. They created an on-ramp for his new wife to be included in the family's deep musical conversations. She needed a low barrier-to-entry, high-ceiling on-ramp to sophisticated music. These are also the motivation behind Bach's Two- and Three-Part Inventions. Bach's life as a composer was inextricably bound up in his life as a teacher, and these accessible works are the "rich problems" of piano education. They are accessible pieces that even the greatest virtuosi find beautiful.

This is why I love Glenn Gould's recordings of Bach's Two- and Three-Part Inventions. They are the music of his childhood, but from an advanced standpoint. The recordings are filled with joy and delight. To the dismay of critics, he often hums along with himself as he plays. The critics find this annoying. Personally, I find it inspiring. If Glenn Gould can still delight in playing these pieces, then so can I. It gives me great permission. The recordings and the playing are magical.

In my view, this is how we should be looking at the balance we need to strike in math education. There are technique and concepts and repertoire that learners need. Without strong technique, your understanding of concepts will be shallow. Without context/repertoire, your understanding and practice of mathematics will be joyless and without wonder.

The long-range purpose
of our practice was clearly
stated on the board
So this is the basis from which I used Kate Nowak's Speed Dating structure to solidify my Precalculus students' early understanding and integration of transformations of functions on Friday. As I keep reminding them, the parent functions and their graphs, as well as the transformations of these basic function graphs, are the essential vocabulary development work for calculus. This is the Hanon and Czerny mindset shift on which we are focusing this year: elementary things that we consider from an advanced standpoint. The order of operations becomes more sophisticated. "Groupings" replace "parentheses" in your thinking about where to start. Groupings, I warn them, are masters of disguise. Sometimes they show up as parentheses, but much of the time they show up wearing a moustache or another costume. They might show up in the form of the absolute value bars or the fraction bar. Don't be fooled, I warn them. Keep your focus on whether you are dealing with transformations of inputs or outputs.

We started simply, using only a single function du jour. On Friday, that was the square root of x. We also restricted our investigation to horizontal and vertical shifts as well as reflections across the x- and y-axis. We were considering the impact on the graphs of basic parent functions as we operated on either the input or the output of the function. We did not multiply by any value other than –1 to start. The Day 1 problem cards are here.  The Day 2 problem cards are also available now (formatted by the amazing Meg Craig - thank you!).

On Friday, each of my 36 students started out with his or her own problem card that contained two related transformations — a shift and a reflection. We organized the speed dating structure, moved our backpacks along the two empty walls, and established our rules of movement (the students along the window side of each row travel, while the hallway-side students stay where they are). If you run into problems, ask the expert in the room on that problem. He or she is sitting directly across from you.
36 precalc students in a state of flow

Trade, analyze, investigate, sketch, discuss. For forty minutes, my Precalculus students lost themselves in analyzing, investigating, sketching, and discussing functional operations on inputs and outputs. Every two minutes, my iPhone timer would go off and I would call out, "Shift!" And all 36 students would trade back their problem cards, while half of them stood up and moved one seat to their right. Then I would reset the timer and they would lose their ego-selves in each new immersion. I loved the hush that fell over the room after everybody settled down into the next round of problems. I eavesdropped on moving and purposeful conversation about inputs and outputs of functions, shifting and reflecting, as they worked collaboratively to help each other and to help themselves attain proficiency. The preciousness of each minute of mathematical conversation was not lost on me.

And I tell you, it was glorious.